UV Image Editor won't allow me to search for textures
Your very top editor is the uv image editor, but to use it you need to drag that whole bar down to see the window - what you think is the uv image editor is actually a node editor, and below that is the 3d view and then the timeline. 

Use ctrl-left/right arrow to get to the UV Editor layout, and then use the button next to the 'New' button to browse that available images in the file - if there is an image texture in the cup's material, then it shoudl be available to open.
[![

edit: your material for the cup doesn't have an image node hooked up, and there isn't an image to go on the cup material in the file. Looks like you need to add an image node and then open the image you say you have in the mtl file, or go back to blender render and try to import the object with the mtl intact to get the image in the material.
Image as reference for the UV Image Editor]1]1